Theorem ablcom 8099

Description: An Abelian group operation is commutative.

Hypothesis

Ref	Expression
ablcom.1	|- X = ran G

Assertion

Ref	Expression
ablcom	|- ((G e. Abel /\ A e. X /\ B e. X) -> (AGB) = (BGA))

Proof of Theorem ablcom

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . 5 |- (x = A -> (xGy) = (AGy))
2		opreq2 3975	. . . . 5 |- (x = A -> (yGx) = (yGA))
3	1, 2	eqeq12d 1492	. . . 4 |- (x = A -> ((xGy) = (yGx) <-> (AGy) = (yGA)))
4		opreq2 3975	. . . . 5 |- (y = B -> (AGy) = (AGB))
5		opreq1 3974	. . . . 5 |- (y = B -> (yGA) = (BGA))
6	4, 5	eqeq12d 1492	. . . 4 |- (y = B -> ((AGy) = (yGA) <-> (AGB) = (BGA)))
7	3, 6	rcla42v 1883	. . 3 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X (xGy) = (yGx) -> (AGB) = (BGA)))
8		ablcom.1	. . . . 5 |- X = ran G
9	8	isabl 8097	. . . 4 |- (G e. Abel <-> (G e. Grp /\ A.x e. X A.y e. X (xGy) = (yGx)))
10	9	pm3.27bi 326	. . 3 |- (G e. Abel -> A.x e. X A.y e. X (xGy) = (yGx))
11	7, 10	syl5com 52	. 2 |- (G e. Abel -> ((A e. X /\ B e. X) -> (AGB) = (BGA)))
12	11	3impib 833	1 |- ((G e. Abel /\ A e. X /\ B e. X) -> (AGB) = (BGA))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  ran crn 3177  (class class class)co 3969  Grpcgr 8030  Abelcabl 8095

This theorem is referenced by:  abl23 8100  ablmuldiv 8103  abldiv23 8106  ghgrpi 8133  ringcom 8149  vccom 8175  nvcom 8236

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-abl 8096

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